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| Mirrors > Home > MPE Home > Th. List > Mathboxes > xlimpnfxnegmnf2 | Structured version Visualization version GIF version | ||
| Description: A sequence converges to +∞ if and only if its negation converges to -∞. (Contributed by Glauco Siliprandi, 23-Apr-2023.) |
| Ref | Expression |
|---|---|
| xlimpnfxnegmnf2.j | ⊢ Ⅎ𝑗𝐹 |
| xlimpnfxnegmnf2.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
| xlimpnfxnegmnf2.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
| xlimpnfxnegmnf2.f | ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) |
| Ref | Expression |
|---|---|
| xlimpnfxnegmnf2 | ⊢ (𝜑 → (𝐹~~>*+∞ ↔ (𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))~~>*-∞)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | xlimpnfxnegmnf2.j | . . 3 ⊢ Ⅎ𝑗𝐹 | |
| 2 | xlimpnfxnegmnf2.z | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
| 3 | xlimpnfxnegmnf2.f | . . 3 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) | |
| 4 | 1, 2, 3 | xlimpnfxnegmnf 45829 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗) ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)-𝑒(𝐹‘𝑗) ≤ 𝑥)) |
| 5 | xlimpnfxnegmnf2.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
| 6 | 1, 5, 2, 3 | xlimpnf 45857 | . 2 ⊢ (𝜑 → (𝐹~~>*+∞ ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗))) |
| 7 | nfmpt1 5250 | . . . 4 ⊢ Ⅎ𝑗(𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗)) | |
| 8 | 3 | ffvelcdmda 7104 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ*) |
| 9 | 8 | xnegcld 13342 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → -𝑒(𝐹‘𝑘) ∈ ℝ*) |
| 10 | nfcv 2905 | . . . . . 6 ⊢ Ⅎ𝑘-𝑒(𝐹‘𝑗) | |
| 11 | nfcv 2905 | . . . . . . . 8 ⊢ Ⅎ𝑗𝑘 | |
| 12 | 1, 11 | nffv 6916 | . . . . . . 7 ⊢ Ⅎ𝑗(𝐹‘𝑘) |
| 13 | 12 | nfxneg 45472 | . . . . . 6 ⊢ Ⅎ𝑗-𝑒(𝐹‘𝑘) |
| 14 | fveq2 6906 | . . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝐹‘𝑗) = (𝐹‘𝑘)) | |
| 15 | 14 | xnegeqd 45448 | . . . . . 6 ⊢ (𝑗 = 𝑘 → -𝑒(𝐹‘𝑗) = -𝑒(𝐹‘𝑘)) |
| 16 | 10, 13, 15 | cbvmpt 5253 | . . . . 5 ⊢ (𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗)) = (𝑘 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑘)) |
| 17 | 9, 16 | fmptd 7134 | . . . 4 ⊢ (𝜑 → (𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗)):𝑍⟶ℝ*) |
| 18 | 7, 5, 2, 17 | xlimmnf 45856 | . . 3 ⊢ (𝜑 → ((𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))~~>*-∞ ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)((𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))‘𝑗) ≤ 𝑥)) |
| 19 | 2 | uztrn2 12897 | . . . . . . 7 ⊢ ((𝑘 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 𝑗 ∈ 𝑍) |
| 20 | xnegex 13250 | . . . . . . . . 9 ⊢ -𝑒(𝐹‘𝑗) ∈ V | |
| 21 | fvmpt4 45244 | . . . . . . . . 9 ⊢ ((𝑗 ∈ 𝑍 ∧ -𝑒(𝐹‘𝑗) ∈ V) → ((𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))‘𝑗) = -𝑒(𝐹‘𝑗)) | |
| 22 | 20, 21 | mpan2 691 | . . . . . . . 8 ⊢ (𝑗 ∈ 𝑍 → ((𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))‘𝑗) = -𝑒(𝐹‘𝑗)) |
| 23 | 22 | breq1d 5153 | . . . . . . 7 ⊢ (𝑗 ∈ 𝑍 → (((𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))‘𝑗) ≤ 𝑥 ↔ -𝑒(𝐹‘𝑗) ≤ 𝑥)) |
| 24 | 19, 23 | syl 17 | . . . . . 6 ⊢ ((𝑘 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (((𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))‘𝑗) ≤ 𝑥 ↔ -𝑒(𝐹‘𝑗) ≤ 𝑥)) |
| 25 | 24 | ralbidva 3176 | . . . . 5 ⊢ (𝑘 ∈ 𝑍 → (∀𝑗 ∈ (ℤ≥‘𝑘)((𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))‘𝑗) ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ≥‘𝑘)-𝑒(𝐹‘𝑗) ≤ 𝑥)) |
| 26 | 25 | rexbiia 3092 | . . . 4 ⊢ (∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)((𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))‘𝑗) ≤ 𝑥 ↔ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)-𝑒(𝐹‘𝑗) ≤ 𝑥) |
| 27 | 26 | ralbii 3093 | . . 3 ⊢ (∀𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)((𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))‘𝑗) ≤ 𝑥 ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)-𝑒(𝐹‘𝑗) ≤ 𝑥) |
| 28 | 18, 27 | bitrdi 287 | . 2 ⊢ (𝜑 → ((𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))~~>*-∞ ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)-𝑒(𝐹‘𝑗) ≤ 𝑥)) |
| 29 | 4, 6, 28 | 3bitr4d 311 | 1 ⊢ (𝜑 → (𝐹~~>*+∞ ↔ (𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))~~>*-∞)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Ⅎwnfc 2890 ∀wral 3061 ∃wrex 3070 Vcvv 3480 class class class wbr 5143 ↦ cmpt 5225 ⟶wf 6557 ‘cfv 6561 ℝcr 11154 +∞cpnf 11292 -∞cmnf 11293 ℝ*cxr 11294 ≤ cle 11296 ℤcz 12613 ℤ≥cuz 12878 -𝑒cxne 13151 ~~>*clsxlim 45833 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-1o 8506 df-2o 8507 df-er 8745 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fi 9451 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-z 12614 df-uz 12879 df-xneg 13154 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-topgen 17488 df-ordt 17546 df-ps 18611 df-tsr 18612 df-top 22900 df-topon 22917 df-bases 22953 df-lm 23237 df-xlim 45834 |
| This theorem is referenced by: (None) |
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